On the Length of Critical Orbits of Stable Quadratic Polynomials (2009)
Ostafe, Alina, Shparlinski, Igor E.
We use the Weil bound of multiplicative character sums together with some recent results of N. Boston and R. Jones, to show that the critical orbit of quadratic polynomials over a finite field of $q$...
Pseudorandom Numbers and Hash Functions from Iterations of Multivariate Polynomials (2009)
Ostafe, Alina, Shparlinski, Igor E.
Dynamical systems generated by iterations of multivariate polynomials with slow degree growth have proved to admit good estimates of exponential sums along their orbits which in turn lead to rather...
Short Cycles in Repeated Exponentiation Modulo a Prime (2009)
Glebsky, Lev, Shparlinski, Igor E.
Given a prime $p$, we consider the dynamical system generated by repeated exponentiations modulo $p$, that is, by the map $u \mapsto f_g(u)$, where $f_g(u) \equiv g^u \pmod p$ and $0 \le f_g(u) \le...
Ahmadi, Omran, Shparlinski, Igor E.
Let $\cC$ be a smooth absolutely irreducible curve of genus $g \ge 1$ defined over $\F_q$, the finite field of $q$ elements, and let $# \cC(\F_{q^n})$ be the number of $\F_{q^n}$-rational points on...
Rank statistics for a family of elliptic curves over a function field (2009)
Pomerance, Carl, Shparlinski, Igor E.
We show that the average and typical ranks in a certain parametric family of elliptic curves described by D. Ulmer tend to infinity as the parameter $d \to\infty$. This is perhaps unexpected since by...
On Point Sets in Vector Spaces over Finite Fields That Determine Only Acute Angle Triangles (2009)
For three points $\vec{u}$,$\vec{v}$ and $\vec{w}$ in the $n$-dimensional space $\F_q^n$ over the finite field $\F_q$ of $q$ elements we give a natural interpretation of an acute angle triangle...
John B. Friedlander, Igor E. Shparlinski
For a given residue class a (mod m) with gcd(a, m) = 1, upper bounds are obtained on the smallest value of n with ϕ(n) ≡ a (mod m). Here, as usual ϕ(n) denotes the Euler function. These bounds...
MULTIPLICATIVE CHARACTER SUMS WITH TWICE-DIFFERENTIABLE FUNCTIONS (2009)
Banks, William D., Shparlinski, Igor E.
For a nontrivial multiplicative character χ modulo p, we bound character sums taken on the integer parts of a real-valued, twice-differentiable function f whose second derivative decays at an...
On the largest prime factor of the Mersenne (2008)
Kevin Ford, Florian Luca, Igor E. Shparlinski
numbers
Distribution of Farey Fractions in Residue Classes and Lang–Trotter Conjectures on Average (2008)
Alina Carmen Cojocaru, Igor E. Shparlinski
We prove that the set of Farey fractions of order T, that is, the set {α/β ∈ Q: gcd(α, β) = 1, 1 � α, β � T}, is uniformly distributed in residue classes modulo a prime p provided T � p...
Divisibility, Smoothness and Cryptographic Applications (2008)
Naccache, David, Shparlinski, Igor E.
This paper deals with products of moderate-size primes, familiarly known as smooth numbers. Smooth numbers play a crucial role in information theory, signal processing and cryptography. We present...
On the Distribution of the Euler Function of Shifted Smooth Numbers (2008)
Loiperdinger, Stefanie S., Shparlinski, Igor E.
We give asymptotic formulas for some average values of the Euler function on shifted smooth numbers. The result is based on various estimates on the distribution of smooth numbers in arithmetic...
On the exponent of the group of points on elliptic curves in extension fields (2008)
Florian Luca, Igor E. Shparlinski
Let E be an elliptic curve defined over Fq, a finite field of q elements. Furthermore, we consider
Classical and Quantum Algorithms for Exponential Congruences (2008)
Van Dam, Wim, Shparlinski, Igor E.
We discuss classical and quantum algorithms for solvability testing and finding integer solutions x,y of equations of the form af^x + bg^y = c over finite fields GF(q). A quantum algorithm with time...
On the uniformity of distribution of the RSA pairs (2008)
Abstract. Let m = pl be a product of two distinct primes p and l. Weshow that for almost all exponents e with gcd(e, ϕ(m)) = 1 the RSA pairs (x, xe) are uniformly distributed modulo m when x runs...
An identification scheme based on sparse polynomials (2008)
William D. Banks, Daniel Lieman, Igor E. Shparlinski
Abstract. This paper gives a new example of exploiting the idea of using polynomials with restricted coefficients over finite fields and rings to construct reliable cryptosystems and identification...
Bounds on the Fourier Coefficients of the Weighted Sum Function (2008)
We estimate Fourier coefficients of a Boolean function which has recently been introduced in the study of read-once branching programs. Our bound implies that this function has an asymptotically...
Harald Niederreiter, Domingo Gómez Pérez, Llorenç Huguet Rotger, Magdalena Payeras Capellà, Luis Pesquera González, Igor E. Shparlinski, ...
3-Loop Networks can have arbitrarily many Minimum Distance Diagrams
Number Theoretic Designs for Directed Regular Graphs of Small Diameter (2008)
William D. Banks, Alessandro Conflitti, Igor E. Shparlinski
In 1989, F. R. K. Chung gave a construction for certain directed h-regular graphs of small diameter. Her construction is based on finite fields, and the upper bound on the diameter of these graphs is...
Constructions of Approximately Mutually Unbiased Bases (2008)
Igor E. Shparlinski, Arne Winterhof
Abstract. We construct systems of bases of C n which are mutually almost orthogonal and which might turn out to be useful for quantum computation. Our constructions are based on bounds of classical...
Elliptic curves with low embedding degree (2008)
Florian Luca, Autónoma México, Igor E. Shparlinski
Motivated by the needs of the pairing based cryptography, Miyaji, Nakabayashi and Takano have suggested a construction of so-called MNT elliptic curves with low embedding degree. We give some...
On the Bit Security of NTRUEncrypt (2008)
Igor E. Shparlinski, William Whyte
Abstract. We show that in certain natural computational models every bit of a message encrypted with the NtruEncrypt cryptosystem is as secure as the whole message.
On the Bit Security of NTRUEncrypt (2008)
Mats Näslund, Igor E. Shparlinski, William Whyte, Ntru Cryptosystems
Abstract. We show that in certain natural computational models every bit of a message encrypted with the NtruEncrypt cryptosystem is as secure as the whole message. 1
Least totient in a residue class (2008)
Friedlander, John B., Shparlinski, Igor E.
We are indebted to Moubariz Garaev, who pointed out a slip in our use of the bound for character sums from [2] in the proof of [1, Corollary 2]. A correct version would be as follows. COROLLARY 2....
Arithmetic properties of Apery numbers (2008)
Luca, Florian, Shparlinski, Igor E.
Let (An)n≥1 be the sequence of Apéry numbers with a general term given by . In this paper, we prove that both the inequalities ω(An) > c0 log log log n and P(An) > c0 (log n log log n)1/2 hold...
Bourgain, Jean, Konyagin, Sergei V., Shparlinski, Igor E.
We give a lower bound on the size of the product set of two arbitrary subsets of the set of Farey fractions of a given order and apply it to study the distribution of elements of multiplicative...
Distribution of Farey fractions in residue classes and Lang-Trotter conjectures on average (2008)
Shparlinski, Igor E, Cojocaru, Alina Carmen
10 page(s)
On the number of sign changes of Hecke eigenvalues of newforms (2008)
Kohnen, Winfried, Lau, Yuk-Yam, Shparlinski, Igor E
We show that, for every x exceeding some explicit bound depending only on k and N, there are at least C(k,N)x/log ¹⁷x positive and negative coefficients a(n) with n≤x in the Fourier expansion of...
Infinite Hilbert class field towers over cyclotomic fields (2008)
We use a result of Y. Furuta to show that for almost all positive integers m, the cyclotomic field Q(exp(2πi/m)) has an infinite Hilbert p-class field tower with high rank Galois groups at each...
Approximation by several rationals (2008)
Following T. H. Chan, we consider the problem of approximation of a given rational fraction a/q by sums of several rational fractions a₁/q₁,…,an/qn with smaller denominators. We show that in...
On the solvability of bilinear equations in finite fields (2008)
Abstract unavailable due to use of unsupported mathematical characters in the formatting.
Graphs with Integral Spectrum (2008)
Omran Ahmadi, Noga Alon, Ian F. Blake, Igor E. Shparlinski
It is shown that only a fraction of 2 −Ω(n) of the graphs on n vertices have an integral spectrum. Although there are several explicit constructions of such graphs, no upper bound for their...
On Pseudosquares and Pseudopowers (2007)
Pomerance, Carl, Shparlinski, Igor E.
Introduced by Kraitchik and Lehmer, an $x$-pseudosquare is a positive integer $n\equiv1\pmod 8$ that is a quadratic residue for each odd prime $p\le x$, yet is not a square. We use bounds of...
On the Distribution of Pseudopowers (2007)
Konyagin, Sergei V., Pomerance, Carl, Shparlinski, Igor E.
An $x$-pseudopower to base $g$ is a positive integer which is not a power of $g$ yet is so modulo $p$ for all primes $p\le x$. We improve an upper bound for the least such number due to E. Bach, R....
On the Convex Closure of the Graph of Modular Inversions (2007)
Khan, Mizan R., Shparlinski, Igor E., Yankov, Christian L.
In this paper we give upper and lower bounds as well as a heuristic estimate on the number of vertices of the convex closure of the set $$ G_n=\left\{(a,b) : a,b\in \Z, ab \equiv 1 \pmod{n}, 1\leq...
David R. Kohel, Igor E. Shparlinski
Abstract. In the paper an upper bound is established for certain exponential sums, analogous to Gaussian sums, dened on the points of an elliptic curve over a prime nite eld. The bound is applied to...
Ecole Normale Sup'erieure (2007)
Phong Q. Nguyen, Igor E. Shparlinski, Jacques Stern
We obtain some uniformity of distribution results for the values of modular sums of the form n
Steven D. Galbraith, Herbie J. Hopkins, Igor E. Shparlinski
Abstract. The Weil and Tate pairings are a popular new gadget in cryptography and have found many applications, including identity-based cryptography. In particular, the pairings have been used for...
Dan Boneh, Igor E. Shparlinski
Let E=F p be an elliptic curve, and G 2 E=F p. Dene the Die{Hellman function on E=F p as DH E;G (aG; bG) = abG. We show that if there is an ecient algorithm for predicting the LSB of the x or y...
On the Security of Di#e--Hellman Bits (2007)
Maria Isabel, Gonzalez Vasco, Igor E. Shparlinski
Abstract. Boneh and Venkatesan have recently proposed a polynomial time algorithm for recovering a "hidden " element # of a finite field IFp of p elements from rather short strings...
Maria Isabel, Gonz Vasco, Igor E. Shparlinski
Boneh and Venkatesan have recently proposed a polynomial time algorithm for recovering a "hidden " element # of a finite field IF p of p elements from rather short strings of the...
Arithmetic and Geometric Progressions in Productsets over Finite Fields (2007)
Given two sets $\cA, \cB \subseteq \F_q$ of elements of the finite field $\F_q$ of $q$ elements, we show that the productset $$ \cA\cB = \{ab | a \in \cA, b \in\cB\} $$ contains an arithmetic...
On RSA Moduli with Almost Half of the Bits Prescribed (2007)
Graham, Sidney W., Shparlinski, Igor E.
We show that using character sum estimates due to H. Iwaniec leads to an improvement of recent results about the distribution and finding RSA moduli $M=pl$, where $p$ and $l$ are primes, with...
On The Solvability of Bilinear Equations in Finite Fields (2007)
We consider the equation $$ ab + cd = \lambda, \qquad a\in A, b \in B, c\in C, d \in D, $$ over a finite field $F_q$ of $q$ elements, with variables from arbitrary sets $ A, B, C, D \subseteq F_q$....
Products in Residue Classes (2007)
Friedlander, John B., Kurlberg, Par, Shparlinski, Igor E.
We consider a problem of P. Erdos, A. M. Odlyzko and A. Sarkozy about the representation of residue classes modulo m by products of two not too large primes. While it seems that even the Extended...
Multiplicative Order of Gauss Periods (2007)
Ahmadi, Omran, Shparlinski, Igor E., Voloch, Jose Felipe
We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of J. von zur Gathen and I. E. Shparlinski.
Approximation by Several Rationals (2007)
Following T. H. Chan, we consider the problem of approximation of a given rational fraction a/q by sums of several rational fractions a_1/q_1, ..., a_n/q_n with smaller denominators. We show that in...
Tate-Shafarevich Groups and Frobenius Fields of Reductions of Elliptic Curves (2007)
Let $\E/\Q$ be a fixed elliptic curve over $\Q$ which does not have complex multiplication. Assuming the Generalized Riemann Hypothesis, A. C. Cojocaru and W. Duke have obtained an asymptotic formula...
Least totient in a residue class (2007)
Friedlander, John B., Shparlinski, Igor E.
For a given residue class a ± od m with gcd(a, m) = 1, upper bounds are obtained on the smallest value of n with ϕ(n) ≡ a ± od m. Here, as usual ϕ(n) denotes the Euler function. These...
On the largest prime factor of the Mersenne numbers (2007)
Ford, Kevin, Luca, Florian, Shparlinski, Igor E.
Let P(k) be the largest prime factor of the positive integer k. In this paper, we prove that the series $\sum_{n\ge 1}\frac{(\log n)^a}{P(2^n-1)}$ is convergent for each constant a
On the distribution of points on multidimensional modular hyperbolas (2007)
We study the distribution of points on the $(n+1)$-dimensional modular hyperbola $a_1\cdots a_{n+1} \equiv c \pmod q$, where $q$ and $c$ are relatively prime integers. In particular, we show that an...
Distribution of Roots of Polynomial Congruences (2007)
For a prime p, we obtain an upper bound on the discrepancy of fractions r/p, where r runs through all of roots modulo p of all monic univariate polynomials of degree d whose vector of coefficients...
Distribution of matrices with restricted entries over finite fields (2007)
Ahmadi, Omran, Shparlinski, Igor E
For a prime p, we consider some natural classes of matrices over a finite field Fp of p elements, such as matrices of given rank or with characteristic polynomial having irreducible divisors of...
Parameters of Integral circulant graphs and periodic quantum dynamics (2007)
Saxena, Nitin, Severini, Simone, Shparlinski, Igor E
The intention of the paper is to move a step towards a classification of network topologies that exhibit periodic quantum dynamics. We show that the evolution of a quantum system whose hamiltonian is...
Estimates for Wieferich numbers (2007)
Banks, William D, Luca, Florian, Shparlinski, Igor E
We define Wieferich numbers to be those odd integers w ≥ 3 that satisfy the congruence 2 φ(w)≡ 1(mod w²). It is clear that the distribution of Wieferich numbers is closely related to the...
Quantum period reconstruction of approximate sequences (2007)
Shparlinski, Igor E, Winterhof, Arne
We consider the problem of determining the period of a sequence over an unknown finite field, given a black-box which returns only a few most significant bits of the sequence elements. For sequences...
On Finite fields for pairing based cryptography (2007)
Luca, Florian, Shparlinski, Igor E
Here, we improve our previous bound on the number of finite fields over which elliptic curves of cryptographic interest with a given embedding degree and small complex multiplication discriminant may...
Exponential sums with polynomial values of the discrete logarithm (2007)
Banks, William D, Shparlinski, Igor E
We estimate exponential sums of the form [equation omitted for formatting reasons] where f is a polynomial with integer cofficients, and ind n is the discrete logarithm of n modulo an odd prime p and...
Integers with a large smooth divisor (2007)
Banks, William D, Shparlinski, Igor E
We study the function O(x, y, z) that counts the number of positive integers n ≤ x which have a divisor d > z with the property that p ≤ y for every prime p dividing d. We also indicate some...
Average distribution of prime ideals in (2007)
Igor E. Shparlinski, José Felipe Voloch
families of number fields
Distribution of Matrices with Restricted Entries over Finite Fields (2007)
Omran Ahmadi, Igor E. Shparlinski
For a prime p, we consider some natural classes of matrices over a finite field Fp of p elements, such as matrices of given rank or with characteristic polynomial having irreducible divisors of...
Visible Points on Curves over Finite (2007)
Igor E. Shparlinski, José Felipe Voloch
For a prime p and an absolutely irreducible modulo p polynomial f(U, V) ∈ Z[U, V] we obtain an asymptotic formulas for the number of solutions to the congruence f(x, y) ≡ a (mod p) in positive...
Voloch, Multiplicative order of Gauss periods (2007)
Omran Ahmadi, Igor E. Shparlinski, José Felipe Voloch
We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of J. von zur Gathen and I. E. Shparlinski....
PARAMETERS OF INTEGRAL CIRCULANT GRAPHS AND PERIODIC QUANTUM DYNAMICS (2007)
Nitin Saxena, Simone Severini, Igor E. Shparlinski
The intention of the paper is to move a step towards a classification of network topologies that exhibit periodic quantum dynamics. We show that the evolution of a quantum system whose hamiltonian is...
Distribution of Roots of Polynomial Congruences (2007)
For a prime p, we obtain an upper bound on the discrepancy of fractions r/p, where r runs through all of roots modulo p of all monic univariate polynomials of degree d whose vector of coefficients...
Bounds on the Fourier coefficients of the weighted sum function (2007)
We estimate Fourier coefficients of a Boolean function which has recently been introduced in the study of read-once branching programs. Our bound implies that this function has rather “flat”...
Bounds of incomplete multiple Kloosterman sums (2007)
We obtain an estimate for incomplete multiple Kloosterman sums modulo a prime which improves the previous result of W. Luo.
Communication complexity of some number theoretic functions (2007)
We show that the communication complexity of the parity of the sum of binary digits of x+y is at least 0.085667...n+O(1) where x and y are n-bit integers. We also obtain a nontrivial (but weaker)...
Chinese remaindering with multiplicative noise (2007)
Shparlinski, Igor E, Steinfeld, Ron
We use lattice reduction to obtain a polynomial-time algorithm for recovering an integer (up to a multiple) given multiples of its residues modulo sufficiently many primes, when the multipliers are...
Distribution of some sequences of points on elliptic curves (2007)
Lange, Tanja, Shparlinski, Igor E
We estimate character sums over points on elliptic curves over a finite field Fq of q elements. Pseudorandom sequences can be constructed by taking linear combinations with small coefficients (for...
Visible points on curve over finite fields (2007)
Shparlinski, Igor E, Voloch, José Felipe
For a prime p and an absolutely irreducible modulo p polynomial f(U, V) E Z[U, V] we obtain an asymptotic formula for the number of solutions to the congruence f(x,y) ≡ a (mod p) in positive...
On the square-free parts of ⌊en!⌋ (2007)
Luca, Florian, Shparlinski, Igor E
In this note, we show that if we write ⌊en!⌋ = s(n)u(n)², where s(n) is square-free then [equation ommitted due to formatting reasons] has at least C log logN distinct prime factors for some...
On values taken by the largest prime factor of shifted primes (2007)
Banks, William D, Shparlinski, Igor E
Let P denote the set of prime numbers, and let P(n) denote the largest prime factor of an integer n > 1. We show that, for every real number 32/17 < η < (4 + 3√2)/4, there exists a constant c(η)...
On the distribution of angles of the Salié sums (2007)
For a prime p and integers a and b, we consider Salié sums [equation omitted for formatting reasons] where χ2(x) is a quadratic character and x¯ is the modular inversion of x, that is, xx¯≡ 1...
On the set of largest prime divisors (2007)
Shparlinski, Igor E, Sutantyo, Daniel
In this paper we obtain lower bounds on the set of the largest prime divisors P(a(n)) of various sequences a(n) for n ≤ x. In particular we obtain such results for polynomial sequences and for...
Least totient in a residue class (2007)
Friedlander, John B, Shparlinski, Igor E
For a given residue class a ± od m with gcd(a, m) = 1, upper bounds are obtained on the smallest value of n with {varphi}(n) ≡ a ± od m. Here, as usual {varphi}(n) denotes the Euler function....
Banks, William D., Shparlinski, Igor E.
We obtain asymptotic formulae for the number of primes $p\le x$ for which the reduction modulo $p$ of the elliptic curve $$ \E_{a,b} : Y^2 = X^3 + aX + b $$ satisfies certain ``natural'' properties,...
Character sums with Beatty sequences on Burgess-type intervals (2006)
Banks, William D., Shparlinski, Igor E.
We estimate multiplicative character sums taken on the values of a non-homogeneous Beatty sequence $\{[\alpha n + \beta] : n =1,2,... \}$, where $\alpha,\beta\in\R$, and $\alpha$ is irrational. Our...
On Some Dynamical Systems in Finite Fields and Residue Rings (2006)
We use character sums to confirm several recent conjectures of V. I. Arnold on the uniformity of distribution properties of a certain dynamical system in a finite field. On the other hand, we show...
Arithmetic properties of the Ramanujan function (2006)
Luca, Florian, Shparlinski, Igor E
We study some arithmetic properties of the Ramanujan function $\tau(n)$, such as the largest prime divisor $P(\tau(n))$ and the number of distinct prime divisors $\omega(\tau(n))$ of $\tau(n)$ for...
Integers with a large smooth divisor (2006)
Banks, William D., Shparlinski, Igor E.
We study the function $\Theta(x,y,z)$ that counts the number of positive integers $n\le x$ which have a divisor $d>z$ with the property that $p\le y$ for every prime $p$ dividing $d$. We also...
Exponential sums with Dickson polynomials (2006)
Gomez-Perez, Domingo, Gutierrez, Jaime, Shparlinski, Igor E
We give new bounds of exponential sums with sequences of iterations of Dickson polynomials over prime finite fields. This result is motivated by possible applications to polynomial generators of...
On the nonlinearity of linear recurrence sequences (2006)
Shparlinski, Igor E, Winterhof, Arne
We obtain an upper bound on exponential sums of a new type with linear recurrence sequences. We apply this bound to estimate the Fourier coefficients, and thus the nonlinearity, of a Boolean function...
On RSA moduli with prescribed bit patterns (2006)
We give a polynomial time probabilistic algorithm that constructs an RSA modulus M=pl, where p and l are two n-bit primes, which has about n/2 bits, on certain positions, prescribed in advance....
On the energy of some circulant graphs (2006)
We give an explicit construction of circulant graphs of very high energy. This construction is based on Gauss sums. We also show the Littlewood conjecture can be used to establish new result for a...
Testing set proportionality and the Ádám isomorphism of circulant graphs (2006)
Coppersmith, Don, Howgrave-Graham, Nick, Nguyễn, Phong Q, Shparlinski, Igor E
Given two k element subsets S,T ⊆ ℤn, we give a quasi-linear algorithm to either find λ ∈ ℤ*n such that S = λT or prove that no such λ exists. This question is closely related to...
Elliptic curves with low embedding degree (2006)
Luca, Florian, Shparlinski, Igor E
Miyaji, Nakabayashi and Takano have recently suggested a construction of the so-called MNT elliptic curves with low embedding degree, which are also of importance for pairing-based cryptography. We...
Distribution of nonlinear congruential pseudorandom numbers modulo almost squarefree integers (2006)
EI-Mahassni, Edwin D, Shparlinski, Igor E, Winterhof, Arne
The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present a new bound on the...
Catalan and Apéry numbers in residue classes (2006)
Garaev, Moubariz Z, Luca, Florian, Shparlinski, Igor E
We estimate character sums with Catalan numbers and middle binomial coefficients modulo a prime p. We use this bound to show that the first at most p13/2(logp)⁶ elements of each sequence already...
Common divisors of the Euler function at related arguments (2006)
Banks, William D, Luca, Florian, Shparlinski, Igor E
Let φ denote the Euler function. For a fixed integer k ≠ 0, we study positive integers n for which the largest prime factor of φ(n) also divides φ(n + k). We obtain an unconditional upper bound...
Small exponent point groups on elliptic curves (2006)
Luca, Florian, Mckee, James, Shparlinski, Igor E
Let E be an elliptic curve defined over Fq, the finite field of q elements. We show that for some constant ƞ > 0 depending only on q, there are infinitely many positive integers n such that the...
Constructions of approximately mutually unbiased bases (2006)
Shparlinski, Igor E, Winterhof, Arne
We construct systems of bases of ℂn which are mutually almost orthogonal and which might turn out to be useful for quantum computation. Our constructions are based on bounds of classical...
On some generalisations of the Erdős Distance Problem over finite fields (2006)
We use exponential sums to obtain new lower bounds on the number of distinct distances defined by all pairs of points (a, b) ∈ A x B for two given sets A, B ⊆ Fnq where Fq is a finite field of q...
Non-residues and primitive roots in Beatty sequences (2006)
Banks, William D, Shparlinski, Igor E
We study multiplicative character sums taken on the values of a non-homogeneous Beatty sequence Bα,β = {⌊αn + β⌋ : n = 1,2,3,…}, where α,β ∈ R, and α is irrational. In particular, our...
Coincidences in the values of the Euler and Carmichael functions (2006)
Banks, William D, Friedlander, John B, Luca, Florian, Pappalardi, Francesco, Shparlinski, Igor E
28 page(s)
On the set of distances between two sets over finite fields (2006)
We use bounds of exponential sums to derive new lower bounds on the number of distinct distances between all pairs of points (x,y)∈ðÂ’œÃ—ℬ for two given...
Character sums and nonlinear recurrence sequences (2006)
Blackburn, Simon R, Shparlinski, Igor E
We obtain upper bounds on character sums and autocorrelation of nonlinear recurrence sequences over arbitrary finite rings.
Arithmetic properties of φ(n)/λ(n) and the structure of the multiplicative group modulo n (2006)
Banks, William D, Luca, Florian, Shparlinski, Igor E
For a positive integer n, we let φ(n) and λ(n) denote the Euler function and the Carmichael function, respectively. We define ξ(n) as the ratio φ(n)/λ(n) and study various arithmetic properties...
GCD of random linear combinations (2006)
Von Zur Gathen, Joachim, Shparlinski, Igor E
We show that for arbitrary positive integers a₁, . . . , am, with probability 6/π² + o(1), the gcd of two linear combinations of these integers with rather small random integer coefficients...
GCD of random linear combinations (2006)
Von Zur Gathen, Joachim, Shparlinski, Igor E
We show that for arbitrary positive integers a₁, . . . , am, with probability 6/π² + o(1), the gcd of two linear combinations of these integers with rather small random integer coefficients...
Double character sums over elliptic curves and finite fields (2006)
Banks, William D, Friedlander, John B, Garaev, Moubariz Z, Shparlinski, Igor E
We estimate certain double character sums over points of an elliptic curve and in the multiplicative subgroup of a finite field. These bounds both improve and extend the scope of a series of previous...
On the sum of iterations of the Euler function (2006)
We study the sum [equation omitted for formating reasons] of consecutive iterations of the Euler function φ(n) (where the last iteration satisfies φ(k(n))(n) = 1). We show that for almost all n,...
On the set of distances between two sets over finite fields (2006)
We use bounds of exponential sums to derive new lower bounds on the number of distinct distances between all pairs of points (x,y) ∈ A×B for two given sets A,B ∈ Fnq, where Fq is a finite field...
Exponential sums with Dickson polynomials (2006)
Gomez-Perez, Domingo, Gutierrez, Jaime, Shparlinski, Igor E
We give new bounds of exponential sums with sequences of iterations of Dickson polynomials over prime finite fields. This result is motivated by possible applications to polynomial generators of...
On the nonlinearity of linear recurrence sequences (2006)
Shparlinski, Igor E, Winterhof, Arne
We obtain an upper bound on exponential sums of a new type with linear recurrence sequences. We apply this bound to estimate the Fourier coefficients, and thus the nonlinearity, of a Boolean function...
On RSA moduli with prescribed bit patterns (2006)
We give a polynomial time probabilistic algorithm that constructs an RSA modulus M=pl, where p and l are two n-bit primes, which has about n/2 bits, on certain positions, prescribed in advance....
On the energy of some circulant graphs (2006)
We give an explicit construction of circulant graphs of very high energy. This construction is based on Gauss sums. We also show the Littlewood conjecture can be used to establish new result for a...
Testing set proportionality and the Ádám isomorphism of circulant graphs (2006)
Coppersmith, Don, Howgrave-Graham, Nick, Nguyễn, Phong Q, Shparlinski, Igor E
Given two k element subsets S,T ⊆ ℤn, we give a quasi-linear algorithm to either find λ ∈ ℤ*n such that S = λT or prove that no such λ exists. This question is closely related to...
Elliptic curves with low embedding degree (2006)
Luca, Florian, Shparlinski, Igor E
Miyaji, Nakabayashi and Takano have recently suggested a construction of the so-called MNT elliptic curves with low embedding degree, which are also of importance for pairing-based cryptography. We...
Distribution of nonlinear congruential pseudorandom numbers modulo almost squarefree integers (2006)
EI-Mahassni, Edwin D, Shparlinski, Igor E, Winterhof, Arne
The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present a new bound on the...
Catalan and Apéry numbers in residue classes (2006)
Garaev, Moubariz Z, Luca, Florian, Shparlinski, Igor E
We estimate character sums with Catalan numbers and middle binomial coefficients modulo a prime p. We use this bound to show that the first at most p13/2(logp)⁶ elements of each sequence already...
Common divisors of the Euler function at related arguments (2006)
Banks, William D, Luca, Florian, Shparlinski, Igor E
Let φ denote the Euler function. For a fixed integer k ≠ 0, we study positive integers n for which the largest prime factor of φ(n) also divides φ(n + k). We obtain an unconditional upper bound...
Small exponent point groups on elliptic curves (2006)
Luca, Florian, Mckee, James, Shparlinski, Igor E
Let E be an elliptic curve defined over Fq, the finite field of q elements. We show that for some constant ƞ > 0 depending only on q, there are infinitely many positive integers n such that the...
Constructions of approximately mutually unbiased bases (2006)
Shparlinski, Igor E, Winterhof, Arne
We construct systems of bases of ℂn which are mutually almost orthogonal and which might turn out to be useful for quantum computation. Our constructions are based on bounds of classical...
On some generalisations of the Erdős Distance Problem over finite fields (2006)
We use exponential sums to obtain new lower bounds on the number of distinct distances defined by all pairs of points (a, b) ∈ A x B for two given sets A, B ⊆ Fnq where Fq is a finite field of q...
Non-residues and primitive roots in Beatty sequences (2006)
Banks, William D, Shparlinski, Igor E
We study multiplicative character sums taken on the values of a non-homogeneous Beatty sequence Bα,β = {⌊αn + β⌋ : n = 1,2,3,…}, where α,β ∈ R, and α is irrational. In particular, our...
Coincidences in the values of the Euler and Carmichael functions (2006)
Banks, William D, Friedlander, John B, Luca, Florian, Pappalardi, Francesco, Shparlinski, Igor E
28 page(s)
Character sums and nonlinear recurrence sequences (2006)
Blackburn, Simon R, Shparlinski, Igor E
We obtain upper bounds on character sums and autocorrelation of nonlinear recurrence sequences over arbitrary finite rings.
Arithmetic properties of φ(n)/λ(n) and the structure of the multiplicative group modulo n (2006)
Banks, William D, Luca, Florian, Shparlinski, Igor E
For a positive integer n, we let φ(n) and λ(n) denote the Euler function and the Carmichael function, respectively. We define ξ(n) as the ratio φ(n)/λ(n) and study various arithmetic properties...
GCD of random linear combinations (2006)
Von Zur Gathen, Joachim, Shparlinski, Igor E
We show that for arbitrary positive integers a₁, . . . , am, with probability 6/π² + o(1), the gcd of two linear combinations of these integers with rather small random integer coefficients...
Distribution of harmonic sums and Bernoulli polynomials modulo a prime (2006)
Garaev, Moubariz Z, Luca, Florian, Shparlinski, Igor E
For a fixed integer s ≥ 1, we estimate exponential sums with harmonic sums [equation omitted for formatting reasons] individually and on average, where Hs (n) is computed modulo a prime p. These...
Arithmetic properties of the Ramanujan function (2006)
Luca, Florian, Shparlinski, Igor E
We study some arithmetic properties of the Ramanujan function τ(n), such as the largest prime divisor P(τ(n)) and the number of distinct prime divisors ω(τ(n)) of τ(n) for various sequences of...
On the bit security of the Diffie-Hellman key (2006)
Blake, Ian F, Garefalakis, Theo, Shparlinski, Igor E
Let Fp be a finite field of p elements, where p is prime. The bit security of the Diffie-Hellman function over subgroups of F*p and of an elliptic curve over Fp, is considered. It is shown that if...
Multiplicative character sums with the sum of g-ary digits function (2006)
Banks, William D, Shparlinski, Igor E
We establish upper bounds for multiplicative character sums with the function σg (n) which computes the sum of the digits of n in a fixed base g ≥ 2. Our results may be viewed as analogues of some...
On the lower bound of the linear complexity over Fp of Sidelnikov sequences (2006)
Garaev, Moubariz Z, Luca, Florian, Shparlinski, Igor E, Winterhof, Arne
For a Sidelnikov sequence of period pm-1, tight lower bounds are obtained on its linear complexity L over Fp. In particular, these bounds imply that, uniformly over all p and m, L is close to its...
Short character sums with Beatty sequences (2006)
Banks, William D, Shparlinski, Igor E
We estimate multiplicative character sums taken on the values of a nonhomogeneous Beatty sequence {⌊αn + β⌋ : n = 1, 2, . . . }, where α, β ∈ R, and α is irrational. In particular, our...
Character sums with exponential functions over smooth numbers (2006)
Banks, William D, Friedlander, John B, Garaev, Moubariz Z, Shparlinski, Igor E
We give nontrivial bounds in various ranges for exponential sums of the form [equation omitted for formatting reasons] and [equation omitted for formatting reasons] where m ≥ 2, ϑ is an element of...
Reconstructing noisy polynomial evaluation in residue rings (2006)
Blackburn, Simon R, Gomez-Perez, Domingo, Gutierrez, Jaime, Shparlinski, Igor E
Let q>1 be an integer and let a and b be elements of the residue ring Zq of integers modulo q. We show how, when given a polynomial f ∈ ℤq[X] and approximations to v₀, v₁ ∈ ℤq such that...
Uniform distribution of some ratios involving the number of prime and integer divisors (2006)
Luca, Florian, Shparlinski, Igor E
We show that the fractional parts of the ratios n/ω(n), n/aω(n), n/τ(n) and n/aτ(n), where a ≥ 2 is a fixed integer and, as usual, ω(n) and τ(n) denote the number of prime divisors and the...
Congruences and rational exponential sums with the Euler function (2006)
Banks, William D, Shparlinski, Igor E
We give upper bounds for the number of solutions to congruences with the Euler function ϕ(n) modulo an integer q ≥ 2. We also give nontrivial bounds for rational exponential sums with ϕ(n)/q.
Average value of the Euler function on binary palindromes (2006)
Banks, William D, Shparlinski, Igor E
We study values of the Euler function ϕ(n) taken on binary palindromes of even length. In particular, if B₂l denotes the set of binary palindromes with precisely 2l binary digits, we derive an...
Primitive points on a modular hyperbola (2006)
For positive integers m, U and V, we obtain an asymptotic formula for the number of integer points (u,v) ∈ [1,U]x[1,V] which belong to the modular hyperbola uv ≡ 1 (mod m) and also have gcd(u,v)...
Uniformity of distribution modulo 1 of the geometric mean prime divisor (2006)
Luca, Florian, Shparlinski, Igor E
We show that the fractional parts of n¹/w⁽ⁿ⁾, n¹/Ω⁽ⁿ⁾ and the geometric mean of the distinct prime factors of n are uniformly distributed modulo 1 as n ranges over all the positive...
On the bit security of the DiffieHellman key (2006)
Ian F. Blake, Theo Garefalakis, Igor E. Shparlinski
Let IFp be a finite field of p elements, where p is prime. The bit security of the Diffie-Hellman function over subgroups of IF ∗ p and of an elliptic curve over IFp, is considered. It is shown...
Bounds on the Fourier Coefficients of the Weighted Sum Function (2006)
We estimate Fourier coefficients of a Boolean function which has recently been introduced in the study of read-once branching programs. Our bound implies that this function has an asymptotically...
On the set of distances between two sets over finite fields (2006)
We use bounds of exponential sums to derive new lower bounds on the number of distinct distances between all pairs of points (x,y)∈𝒜×ℬ for two given sets 𝒜,ℬ∈Fqn, where Fq is a finite...
Incomplete exponential sums and Diffie-Hellman triples (2006)
Banks, William D, Friedlander, John B, Konyagin, Sergei V, Shparlinski, Igor E
Let p be a prime and ϑ an integer of order t in the multiplicative group modulo p. In this paper, we continue the study of the distribution of Diffie–Hellman triples (ϑx, ϑy, ϑxy ) by...
Truncations of L-functions in residue classes (2006)
Let χ(n) be a quadratic character modulo a prime p. For a fixed integer s ≠ 0, we estimate certain exponential sums with truncated L-functions [equation removed for for formatting reasons]. Our...
Complexity of inverting the Euler function (2006)
Contini, Scott, Croot, Ernie, Shparlinski, Igor E
Given an integer n, how hard is it to find the set of all integers m such that φ(m) = n, where φ is the Euler totient function? We present a certain basic algorithm which, given the prime number...
Uniform Distribution of Fractional Parts Related to Pseudoprimes (2005)
Banks, William D., Garaev, Moubariz Z., Luca, Florian, Shparlinski, Igor E.
We estimate exponential sums with the Fermat-like quotients $$ f_g(n) = \frac{g^{n-1} - 1}{n} \mand h_g(n)=\frac{g^{n-1}-1}{P(n)}, $$ where $g$ and $n$ are positive integers, $n$ is composite, and...
Prime divisors of some shifted products (2005)
Eric Levieil, Florian Luca, Igor E. Shparlinski
We study prime divisors of various sequences of positive integers A(n)+1, n=1,…,N, such that the ratios a(n)=A(n)/A(n−1) have some number-theoretic or combinatorial meaning....
On the average value of divisor sums in arithmetic progressions (2005)
Banks, William D, Heath-Brown, Roger, Shparlinski, Igor E
We consider very short sums of the divisor function in arithmetic progressions prime to a fixed modulus and show that "on average" these sums are close to the expected value. We also give...
On the average value of divisor sums in arithmetic progressions (2005)
Banks, William D, Heath-Brown, Roger, Shparlinski, Igor E
We consider very short sums of the divisor function in arithmetic progressions prime to a fixed modulus and show that "on average" these sums are close to the expected value. We also give...
On the average value of divisor sums in arithmetic progressions (2005)
Banks, William D, Heath-Brown, Roger, Shparlinski, Igor E
We consider very short sums of the divisor function in arithmetic progressions prime to a fixed modulus and show that "on average" these sums are close to the expected value. We also give...
On the average value of divisor sums in arithmetic progressions (2005)
Banks, William D., Heath-Brown, Roger, Shparlinski, Igor E.
We consider very short sums of the divisor function in arithmetic progressions prime to a fixed modulus and show that “on average” these sums are close to the expected value. We also give...
Values of arithmetical functions equal to a sum of two squares (2005)
Banks, William D., Luca, Florian, Saidak, Filip, Shparlinski, Igor E.
Let ϕ(n) denote the Euler function. In this paper, we determine the order of growth for the number of positive integers n≤x for which ϕ(n) is the sum of two square numbers. We also obtain...
Prime divisors of some shifted products (2005)
Eric Levieil, Florian Luca, Igor E. Shparlinski
We study prime divisors of various sequences of positive integers A(n)+1, n=1,…,N, such that the ratios a(n)=A(n)/A(n−1) have some number-theoretic or combinatorial meaning. In the case a(n)=n,...
Values of arithmetical functions equal to a sum of two squares (2005)
Banks, William D., Luca, Florian, Saidak, Filip, Shparlinski, Igor E.
Let ϕ(n) denote the Euler function. In this paper, we determine the order of growth for the number of positive integers n≤x for which ϕ(n) is the sum of two square numbers. We also obtain...
Prime divisors of sequences associated to elliptic curves (2004)
Everest, Graham, Shparlinski, Igor E
We consider the primes which divide the denominator of the x-coordinate of a sequence of rational points on an elliptic curve. It is expected that for every sufficiently large value of the index,...
Character Sums and Congruences with n! (2004)
Garaev, Moubariz Z., Luca, Florian, Shparlinski, Igor E.
We estimate character sums with n!, on average, and individually. These bounds are used to derive new results about various congruences modulo a prime p and obtain new information about the spacings...
Exponential Sums and Congruences with Factorials (2004)
Garaev, Moubariz Z., Luca, Florian, Shparlinski, Igor E.
We estimate the number of solutions of certain diagonal congruences involving factorials. We use these results to bound exponential sums with products of two factorials $n!m!$ and also derive...
Exponential sums over Mersenne numbers (2004)
Banks, William D., Conflitti, Alessandro, Friedlander, John B., Shparlinski, Igor E.
Average order in cyclic groups (2004)
Von Zur Gathen, Joachim, Knopfmacher, Arnold, Luca, Florian, Lucht, Lutz G., Shparlinski, Igor E.
Exponential sums over Mersenne numbers (2004)
Banks, William D, Conflitti, Alessandro, Friedlander, John B, Shparlinski, Igor E
We give estimates for exponential sums of the form ∑n≤N Λ(n) exp (2πiagⁿ/m), where m is a positive integer, a and g are integers relatively prime to m, and Λ is the von Mangoldt function. In...
BOUNDS OF GAUSS SUMS IN FINITE FIELDS (2004)
Abstract. We consider Gauss sums of the form Gn(a) = � χ(x n) x∈F p m with a nontrivial additive character χ � = χ0 of a finite field Fp m of pm elements of characteristic p. The classical...
Randolph E. Bank, Christine Bernardi, Peter B. Borwein, David W. Boyd, Susanne C. Brenner, Richard P. Brent, ...
Available electronically at www.ams.org/mcom/ Mathematics of Computation This journal is devoted to research articles of the highest quality in computational mathematics. Areas covered include...
Randolph E. Bank, Christine Bernardi, David W. Boyd, Susanne C. Brenner, Richard P. Brent, Joe P. Buhler, ...
Available electronically at www.ams.org/mcom/ Mathematics of Computation This journal is devoted to research articles of the highest quality in computational mathematics. Areas covered include...
On Reducing a System of Equations to a Single Equation (2004)
Gudmund Skovbjerg Frandsen, Igor E. Shparlinski
For a system of polynomial equations over Q p we present an e#cient construction of a single polynomial of quite small degree whose zero set over Q p coincides with the zero set over Q p of the...
Randolph E. Bank, Christine Bernardi, David W. Boyd, Susanne C. Brenner, Richard P. Brent, Joe P. Buhler, ...
Available electronically at www.ams.org/mcom/ Mathematics of Computation This journal is devoted to research articles of the highest quality in computational mathematics. Areas covered include...
Randolph E. Bank, Christine Bernardi, Peter B. Borwein, David W. Boyd, Susanne C. Brenner, Richard P. Brent, ...
Available electronically at www.ams.org/mcom/ Mathematics of Computation This journal is devoted to research articles of the highest quality in computational mathematics. Areas covered include...
Randolph E. Bank, David W. Boyd, Susanne C. Brenner, Richard P. Brent, Joe P. Buhler, Carsten Carstensen, ...
Available electronically at www.ams.org/mcom/ Mathematics of Computation This journal publishes research articles in computational mathematics. Areas covered include numerical analysis, with emphasis...
Randolph E. Bank, Christine Bernardi, David W. Boyd, Susanne C. Brenner, Richard P. Brent, Joe P. Buhler, ...
Available electronically at www.ams.org/mcom/ Mathematics of Computation This journal publishes research articles in computational mathematics. Areas covered include numerical analysis, with emphasis...
Randolph E. Bank, David W. Boyd, Susanne C. Brenner, Richard P. Brent, Joe P. Buhler, Carsten Carstensen, ...
Available electronically at www.ams.org/mcom/ Mathematics of Computation This journal publishes research articles in computational mathematics. Areas covered include numerical analysis, with emphasis...
Randolph E. Bank, Christine Bernardi, David W. Boyd, Susanne C. Brenner, Richard P. Brent, Joe P. Buhler, ...
Available electronically at www.ams.org/mcom/ Mathematics of Computation This journal is devoted to research articles of the highest quality in computational mathematics. Areas covered include...
The hidden number problem with the trace and bit security of (2002)
Mats Näslund, Igor E. Shparlinski
Abstract. We consider a certain generalization of the hidden number problem introduced by Boneh and Venkatesan in 1996. Considering the XTR variation of Diffie-Hellman, we apply our results to show...
Selective forgery of rsa signatures with fixedpattern padding (2002)
Arjen K. Lenstra, Igor E. Shparlinski
Abstract. We present a practical selective forgery attack against RSA signatures with fixed-pattern padding shorter than two thirds of the modulus length. Our result extends the practical existential...
The hidden number problem with the trace and bit security of (2002)
Mats Näslund, Igor E. Shparlinski
Abstract. We consider a certain generalization of the hidden number problem introduced by Boneh and Venkatesan in 1996. Considering the XTR variation of Diffie-Hellman, we apply our results to show...
Instituto de Matematicas (2002)
Arnold Knopfmacher, Florian Luca, Autonoma Mexico, Lutz G. Lucht, Igor E. Shparlinski
For each natural number n we determine the average order #(n) of the elements in a cyclic group of order n. We show that a large fraction of the contribution to #(n) comes from the #(n) primitive...
On the Hardness of Approximating the Permanent of Structured Matrices (2002)
Bruno Codenotti, Igor E. Shparlinski, Arne Winterhof
We show that for several natural classes of "structured" matrices, including symmetric, circulant, Hankel and Toeplitz matrices, approximating the permanent modulo a prime p is as hard as...
Secure Bilinear Diffie-Hellman Bits (2002)
Steven D. Galbraith, Herbie J. Hopkins, Igor E. Shparlinski
Abstract. The Weil and Tate pairings are a popular new gadget in cryptography and have found many applications, including identity-based cryptography. In particular, the pairings have been used for...
A variant of NTRU with non-invertible polynomials (2002)
William D. Banks, Igor E. Shparlinski
Abstract. We introduce a generalization of the NTRU cryptosystem and describe its advantages and disadvantages as compared with the original NTRU protocol. This extension helps to avoid the potential...
On the insecurity of a server-aided RSA protocol (2001)
Phong Q. Nguyen, Igor E. Shparlinski
Abstract. At Crypto ’88, Matsumoto, Kato and Imai proposed a protocol, known as RSA-S1, in which a smart card computes an RSA signature, with the help of an untrusted powerful server. There exist...
On the unpredictability of bits of the elliptic curve Diffie–Hellman scheme (2001)
Dan Boneh, Igor E. Shparlinski
Abstract. Let E/Fp be an elliptic curve, and G ∈ E/Fp. Define the Diffie–Hellman function as DHE,G(aG, bG) = abG. We show that if there is an efficient algorithm for predicting the LSB of the x...
On the security of Lenstra’s variant of DSA without long inversions (2001)
Arjen K. Lenstra, Igor E. Shparlinski
Abstract. We use bounds of exponential sums to show that for a wide class of parameters the modification of the DSA signature scheme proposed by A. K. Lenstra at Asiacrypt’96 is as secure as the...
Thuong To, \Cryptographic Applications of Sparse Polynomials over Finite Rings (2000)
William D. Banks, Daniel Lieman, Igor E. Shparlinski, Van Thuong To
Abstract. This paper gives new examples that exploit the idea of using sparse polynomials with restricted coefficients over a finite ring for designing fast, reliable cryptosystems and identification...
Security of polynomial transformations of the Di#e--Hellman key (2000)
D. Boneh and R. Venkatesan have recently proposed an approach to proving that a reasonably small portions of most significant bits of the Di#e--Hellman key modulo a prime are as secure the the whole...
The Insecurity of the Digital Signature Algorithm with Partially Known Nonces (2000)
Phong Q. Nguyen, Igor E. Shparlinski
. We present a polynomial-time algorithm that provably recovers the signer's secret DSA key when a few bits of the random nonces k (used at each signature generation) are known for a number of...
On the linear complexity of the Naor-- Reingold pseudo-random function (1999)
William D. Banks, Frances Griffin, Daniel Lieman, Igor E. Shparlinski
Abstract. We obtain an exponential lower bound on the non-linear complexity of the new pseudo-random function, introduced recently by M. Naor and O. Reingold. This bound is an extension of the lower...
On polynomial representations of Boolean functions related to some number theoretic problems (1998)
Erion Plaku, Igor E. Shparlinski
Abstract. We say a polynomial P over ZM strongly M-represents a Boolean function F if F(x) ≡ P(x) (mod M) for all x ∈ {0, 1} n. Similarly, P one-sidedly M-represents F if F(x) = 0 ⇐ ⇒ P(x)...
Computational and algorithmic in finite fields / by Igor E. Shparlinski (1992)
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Computational and algorithmic problems in finite fields / by Igor E. Shparlinski (1992)
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This document in subdirectoryRS/04/6/ On Reducing a System of Equations to a Single Equation
Gudmund Skovbjerg Frandsen, Igor E. Shparlinski, Copyright C, Gudmund Skovbjerg Fr, Igor E, Gudmund S. Frandsen, ...
Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS...